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Spherical varieties and non-ordinary families of cohomology classes

Rob Rockwood

TL;DR

The paper develops a non-ordinary, finite-slope extension of Loeffler’s spherical-pair machinery to construct p-adic families of cohomology classes on locally symmetric spaces and to map them into Galois cohomology via Abel–Jacobi maps. It introduces locally analytic and Iwasawa function spaces, slope decompositions, and control theorems to enable interpolation across weight space, while preserving norm relations through analytic branching maps and congruences. The approach yields distribution-valued Galois cohomology classes and a robust framework for p-adic Euler systems and p-adic L-functions beyond the ordinary locus, with explicit application to GSp$_4$ and the Lemma–Flach Euler system in families. These results broaden the reach of Euler-system techniques to finite-slope settings and offer new tools for studying p-adic variation of automorphic Galois representations. The work provides a concrete pathway to interpolating Beilinson–Eisenstein and Lemma–Flach–type classes across Coleman families, potentially yielding new p-adic L-functions associated to Siegel modular forms and related objects.

Abstract

We give a construction of non-ordinary $p$-adic families of classes in the cohomology of locally symmetric spaces associated to spherical pairs of reductive groups. In the étale case, we show how to map these classes into Galois cohomology. The methods developed in this paper can be used to give new constructions of $p$-adic families of Euler systems and $p$-adic $L$-functions. As an example, we show how the constructions of this paper can be used to construct norm-compatible classes associated to non-ordinary Siegel modular forms, generalising $p$-part of the Lemma--Flach Euler system constructed by Loeffler--Skinner--Zerbes.

Spherical varieties and non-ordinary families of cohomology classes

TL;DR

The paper develops a non-ordinary, finite-slope extension of Loeffler’s spherical-pair machinery to construct p-adic families of cohomology classes on locally symmetric spaces and to map them into Galois cohomology via Abel–Jacobi maps. It introduces locally analytic and Iwasawa function spaces, slope decompositions, and control theorems to enable interpolation across weight space, while preserving norm relations through analytic branching maps and congruences. The approach yields distribution-valued Galois cohomology classes and a robust framework for p-adic Euler systems and p-adic L-functions beyond the ordinary locus, with explicit application to GSp and the Lemma–Flach Euler system in families. These results broaden the reach of Euler-system techniques to finite-slope settings and offer new tools for studying p-adic variation of automorphic Galois representations. The work provides a concrete pathway to interpolating Beilinson–Eisenstein and Lemma–Flach–type classes across Coleman families, potentially yielding new p-adic L-functions associated to Siegel modular forms and related objects.

Abstract

We give a construction of non-ordinary -adic families of classes in the cohomology of locally symmetric spaces associated to spherical pairs of reductive groups. In the étale case, we show how to map these classes into Galois cohomology. The methods developed in this paper can be used to give new constructions of -adic families of Euler systems and -adic -functions. As an example, we show how the constructions of this paper can be used to construct norm-compatible classes associated to non-ordinary Siegel modular forms, generalising -part of the Lemma--Flach Euler system constructed by Loeffler--Skinner--Zerbes.
Paper Structure (32 sections, 47 theorems, 291 equations)

This paper contains 32 sections, 47 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Algebraic representations
  4. Integral lattices
  5. Cohomology of locally symmetric spaces
  6. Branching laws for algebraic representations
  7. Weight spaces
  8. Hecke algebras
  9. Locally analytic function spaces
  10. Locally Iwasawa functions
  11. Locally analytic branching laws
  12. Analytic branching maps
  13. Complexes of Banach modules
  14. Slope decompositions
  15. Slope decompositions on cohomology
  16. ...and 17 more sections

Key Result

Proposition 2.5

Let $\lambda \in X^{\bullet}_{+}(S_{G})^{Q_{H}^{0}}$ and let $\mu \in X_{+}^{\bullet}(S_{H})$ be the associated weight of $H$. Then there is a unique $H$-equivariant map sending $\delta_{1}$ to $f^{\mathrm{sph}}_{\lambda}$.

Theorems & Definitions (139)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 129 more